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Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm. (English) Zbl 0931.76059
We develop a variant of the wave-propagation algorithm which addresses the title problem by introducing a Riemann problem at the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems give high-resolution results. The one- and two-dimensional shallow water equations for flow over arbitrary bottom topography are used as an example, through the ideas apply to many other systems. The method is easily implemented in the software package CLAWPACK. \(\copyright\) Academic Press.

76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76L05 Shock waves and blast waves in fluid mechanics
35L65 Hyperbolic conservation laws
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[1] Bermudez, A.; Vazquez, M., Upwind methods for hyperbolic conservations laws with source term, Comput. fluids, 23, 1049, (1994) · Zbl 0816.76052
[2] Eulderink, R.; Mellema, G., General relativistic hydrodynamics with a roe solver, Astron. astrophys. suppl. ser., 110, 587, (1995)
[3] Falcovitz, J.; Ben-Artzi, M., Recent developments of the GRP method, JSME int. J. ser. B, 38, 497, (1995)
[4] Glaz, H.M.; Liu, T.-P., The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow, Adv. appl. math., 5, 111, (1984) · Zbl 0598.76065
[5] Glimm, J.; Marshall, G.; Plohr, B., A generalized Riemann problem for quasi-one-dimensional gas flows, Adv. appl. math., 5, 1, (1984) · Zbl 0566.76056
[6] Godlewski, E.; Raviart, P.-A., Numerical approximation of hyperbolic systems of conservation laws, (1996) · Zbl 1063.65080
[7] Greenberg, J.M.; LeRoux, A.Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. numer. anal., 33, 1, (1996)
[8] Greenberg, J.M.; LeRoux, A.Y.; Baraille, R.; Noussair, A., Analysis and approximation of conservation laws with source terms, SIAM J. numer. anal., 34, 1980, (1997) · Zbl 0888.65100
[9] Houghton, D.D.; Kasahara, A., Nonlinear shallow fluid flow over an isolated ridge, Comm. pure appl. math., 21, 1, (1968) · Zbl 0159.57404
[10] Jenny, P., On the numerical solution of the compressible navier – stokes equations for reacting and non-reacting gas mixtures, (1997)
[11] Jenny, P.; Müller, B., A new approach for a flux solver taking into account source terms, viscous and multidimensional effects, Proc. 7th intl. conf. on hyperbolic problems, (1998)
[12] P. Jenny, B. Müller, Rankine-Hugoniot-Riemann solver considering source terms and multidimensional effects · Zbl 0926.76079
[13] Kröner, D., Numerical schemes for conservation laws, (1997) · Zbl 0872.76001
[14] R. J. LeVeque, {\scclawpack}, , ,
[15] LeVeque, R.J., Intermediate boundary conditions for time-split methods applied to hyperbolic partial differential equations, Math. comput., 47, 37, (1986) · Zbl 0596.65062
[16] LeVeque, R.J., Numerical methods for conservation laws, (1990) · Zbl 0682.76053
[17] LeVeque, R.J., Wave propagation algorithms for multi-dimensional hyperbolic systems, J. comput. phys., 131, 327, (1997) · Zbl 0872.76075
[18] R. J. LeVeque, D. S. Bale, Wave-propagation methods for conservation laws with source terms, Proc. 7th Intl. Conf. on Hyperbolic Problems, R. Jeltsch, 1998 · Zbl 0927.35062
[19] R. J. LeVeque, D. Mihalas, E. Dorfi, E. Müller, Numerical Methods in Astrophysical Fluid Flow, Saas-Fee Advanced Course 27, A. GautschyO. Steiner, Springer-Verlag, Berlin/New York,
[20] Mellema, G.; Eulderink, F.; Icke, V., Hydrodynamical models of aspherical planetary nebulae, Astron. astrophys., 252, 718, (1991)
[21] K. Murawski, 1998
[22] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[23] P. L. Roe, Upwind differencing schemes for hyperbolic conservation laws with source terms, Nonlinear Hyperbolic Problems, C. CarrasoP.-A. RaviartD. Serre, 1270, Springer-Verlag, Berlin/New York, 1986, 41 · Zbl 0626.65086
[24] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997) · Zbl 0888.76001
[25] van Leer, B., On the relation between the upwind-differencing schemes of Godunov, engquist-osher, and roe, SIAM J. sci. stat. comput., 5, 1, (1984) · Zbl 0547.65065
[26] G. Watson, D. H. Peregrine, E. F. Toro, Numerical solution of the shallow-water equations on a beach using the weighted average flux method, Computational Fluid Dynamics ’92, C. Hirsch, et al. Elsevier, Amsterdam/New York, 1992, 495
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